How do I smooth hexagonal shapes more into cylinders?
In edit mode, use Ctrl-R to add loop cuts to the sides of the 6 sided cylinder. Once you have an even division, select the vertices of the cap or end hexagon. Using Loop Tools addon, choose LoopTools> Circle, and the new vertices will form a circle. Repeat for the other hexagon.
edit:
Here is how I would approach it.
1. Original 2. Tris to quads 3. Delete the vert in center of the hexagon 4. ctrl-R loop cut the faces 5. LoopTools Circle to get the round 6. Extrude and carefully fill with quads